1. **Problem Statement:** We are given the graphs of the derivatives $f'(x)$ and $g'(x)$ and asked to determine which graph could represent the function $h(x) = f(x) - g(x)$. 2. **Key Concept:** The derivative of $h(x)$ is given by $$ h'(x) = f'(x) - g'(x). $$ This means the graph of $h'(x)$ is the vertical difference between the graphs of $f'(x)$ and $g'(x)$ at each $x$. 3. **Given:** - $f'(x)$ is a ray starting at the origin and pointing upwards and to the right, so $f'(x)$ is increasing and positive for $x>0$. - $g'(x)$ is a ray along the positive $x$-axis, so $g'(x) = 0$ for all $x$. 4. **Calculate $h'(x)$:** Since $g'(x) = 0$, we have $$ h'(x) = f'(x) - 0 = f'(x). $$ Therefore, $h'(x)$ has the same shape as $f'(x)$. 5. **Interpretation:** - $h'(x)$ is increasing and positive for $x>0$. - Integrating $h'(x)$ to get $h(x)$, since $h'(x)$ is increasing and positive, $h(x)$ will be a function that is increasing and concave up. 6. **Matching with options:** - Option (a): Parabola opening upwards with vertex at origin — this matches an increasing, concave up function starting at zero. - Option (b): Increasing linear function — derivative would be constant, but $h'(x)$ is increasing. - Option (c): Parabola opening upwards but vertex not at origin — possible but less likely since $h'(0) = f'(0) - g'(0) = 0$ implies vertex at origin. - Option (d): Horizontal line — derivative zero, contradicts $h'(x)$ increasing. **Final answer:** The graph of $h(x)$ is best represented by option (a). $$\boxed{\text{Option (a)}}$$